Fick-Jacobs equation for channels over three-dimensional curves

The purpose of this paper is to provide a new formula for the effective diffusion coefficient of a generalized Fick-Jacobs equation for narrow three-dimensional channels. The generalized Fick-Jacobs equation is obtained by projecting the three-dimensional diffusion equation along the normal directions of a curve in three-dimensional space that roughly resembles the narrow channel. The projection (or dimensional reduction) is achieved by integrating the diffusion equation along the cross sections of the channel contained in the planes orthogonal to the curve. We show that the resulting formula for the associated effective diffusion coefficient can be expressed in terms of the geometric moments of the channel's cross sections and the curve's curvature. We show the effect that a rotating cross section with offset has on the effective diffusion coefficient.

Figure 1

Region ${\Omega }_u$ and cross section $S_u$.

Figure 2

Average orientation and sizes of a channel's cross section with respect to $\langle \mathit{\phi }\rangle$.

Figure 3

A twisted elliptical channel over a helix (left) and the corresponding effective diffusion coefficient function (right). The relevant parameters are $D=1,a=1/4,b=1/6,r_1=1/6,r_2=1/10,p=0,q=0,$ and $\omega =4$.

Figure 4

Geometric series approximation to the effective diffusion coefficient for a channel with gyrating elliptical cross section. The parameters used to generate these curves are the same as those in Fig. 3.

Figure 5

A twisted channel with rectangular cross section. The parameters used to generate the channel are $a=1/4,b=1/6,d_1=1/6,d_2=1/10,p=0,q=0$, and $\omega =4$.

Figure 6

A twisted channel with cardioidal cross section. The parameters used to generate this channel are $a=1/4,b=1/6,r=1/25,\omega =4,p=0,\text{and}q=0$.

Figure 7

Elliptical, rectangular, and cardioidal sections having the same width values $s_1$ and $s_2$.

Figure 8

Pairwise comparisons of the effective diffusion coefficients for the twisted elliptical, rectangular, and cardioidal channels with $r=1/20$.

Figure 9

Pairwise comparisons of the effective diffusion coefficients for the twisted elliptical, rectangular, and cardioidal channels with $r=1/15$.